Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{a^2 - 9}{a - 3}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = a$ $ b = \sqrt{9} = -3$ So we can rewrite the expression as: $z = \dfrac{({a} {-3})({a} + {3})} {a - 3} $ We can divide the numerator and denominator by $(a - 3)$ on condition that $a \neq 3$ Therefore $z = a + 3; a \neq 3$